I would like to be able to use the password associated with this account to install packages with the GUI. Ideally, I would change the settings so that any user with sudo permissions could install packages. If that is not possible, I would like to be able to list all users that should be able to install. I know that this involves making changes to Policy Kit, and I have read a page with a similar question. However, I must need information specific to my own problem, because all my attempts so far have failed.
If, when you say 'this account', you mean your own one then you don't need to change anything because what you're describing is the default behavior. I've just set up a 'standard' account on my own system to check and it's working fine. If you want people to install with their own password you just need to give them administrative privileges.